Express arbitrary operator in terms of x and p

I believe to have read somewhere (but now I can't find it) that any operator in QM is expressible in terms of the x and p operators. Obviously the spin operator is not, so this is not entirely true. But can we say something along those lines? Which operators are expressible in terms of x and p?

I'm thinking about this because in Dirac's original paper on the Dirac equation, he says that the alpha operators(matrices) are not functions of either x nor p, which makes sense. Can we conclude something about the alpha operators based on this?